$2$-large sets are sets of nil-Bohr recurrence

Ryan Alweiss; Sohail Farhangi

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arXiv · 2610.16605

$2$-large sets are sets of nil-Bohr recurrence

Ryan Alweiss·Sohail Farhangi

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Abstract

A set $S$ of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in $S$. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood $B$ of $0$ and every $\eps>0$, restricting the common difference to $B$ increases the extremal number for $k$-term arithmetic progressions by at most a factor of $1+\eps$, provided $k$ is sufficiently large.

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Ryan Alweiss, Sohail Farhangi. 2026-10-12. $2$-large sets are sets of nil-Bohr recurrence. https://arxiv.org/abs/2610.16605

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